Download Liquidity, Business Cycles and Monetary Policy

Liquidity, Business Cycles, and Monetary
Policy
Nobuhiro Kiyotaki and John Moorey
First version, June 2001
This version, February 2012
Abstract
The paper presents a model of a monetary economy where there
are di¤erences in liquidity across assets. Money circulates because it is
more liquid than other assets, not because it has any special function.
There is a spectrum of returns on assets, re‡ecting their di¤erences
in liquidity. The model is used, …rst, to investigate how aggregate
activity and asset prices ‡uctuate with shocks to productivity and
liquidity; second, to examine what role government policy might have
through open market operations that change the mix of assets held by
the private sector. With its emphasis on liquidity rather than sticky
prices, the model harks back to an earlier interpretation of Keynes
(1936), following Tobin (1969).
JEL Classi…cation: E10, E44, E50
The …rst version of this paper was presented in June 2001 as a plenary address to
the Annual Meeting of the Society for Economic Dynamics held in Stockholm; then in
November 2001 as a Clarendon Lecture at the University of Oxford (Kiyotaki and Moore,
2001). We are grateful for feedback from many conference and seminar participants. We
would like to express our special gratitude to Wei Cui for his excellent research assistance.
y
Kiyotaki: Princeton University. Moore: Edinburgh University and London School of
Economics.
1
1
Introduction
This paper presents a model of a monetary economy where there are di¤erences in liquidity across assets. Our aim is to study how aggregate activity
and asset prices ‡uctuate with recurrent shocks to productivity and liquidity.
In doing so, we examine what role government policy might have through
open market operations that change the mix of assets held by the private
sector.
Part of our purpose is to construct a workhorse model of money and
liquidity that does not stray too far from the other workhorse of modern
macroeconomics, the real business cycle model. We thus maintain the assumption of competitive markets. In a standard competitive framework,
money has no role unless endowed with a special function, for example that
the purchase of goods requires cash in advance. In our model, the reason why money can improve resource allocation is not because money has
a special function but because, crucially, we assume that other assets are
partially illiquid, less liquid than money. Ours might be thought of as a
liquidity-in-advance framework.
Iliquidity has to do with some impediment to the resale of assets. With
this in mind, we construct a model in which the resale of assets is a central
feature of the economy. We consider a group of entrepreneurs, who each uses
his or her own capital stock and skill to produce output from labor (which is
supplied by workers). Capital depreciates and is restocked through investment, but the investment technology, for producing new capital from output,
is not commonly available: in each period only some of the entrepreneurs are
able to invest, and the arrival of investment opportunities is randomly distributed across entrepreneurs through time. Hence in each period there is
a need to channel output from those entrepreneurs who don’t have an investment opportunity (that period’s savers) to those who do (that period’s
investors).
To acquire output for the production of new capital, an investing entrepreneur issues equity claims to the capital’s future returns. However, we
assume that because the investing entrepreneur’s skill will be needed to produce these future returns and he cannot precommit to work, at the time of
investment he can credibly pledge only a fraction – say – of the future
returns from the new capital. Unless is high enough, he faces a borrowing
constraint: he must …nance part of the cost of investment from his available
resources. The lower is ; the tighter is the borrowing constraint: the larger
2
is the downpayment per unit of investment that he must make out of his own
funds.
He will typically have on his balance sheet two kinds of asset that can
be resold to raise funds. He may have money. And he may have equity
previously issued by other entrepreneurs. Both of these will have been
acquired by him at some point in the past, when he himself was a saver.
Crucially, we suppose that equity is less liquid than money. We parameterize the degree to which equity is illiquid by making a stylized assumption:
in each period only a proportion –say –of an agent’s equity holding can
be resold. Think of peeling an onion slowly, layer by layer, a fraction
per period. Although the entrepreneur with an investment opportunity this
period can readily divest of his equity holding, to divest any more he will
have to wait until next period, by which time the opportunity may have disappeared. The lower is ; the tighter is the resaleability constraint. Unlike
his equity holding, the entrepreneur’s money holding is perfectly liquid: it
can all be used to buy goods straightaway.
In practice, of course, there are wide di¤erences in resaleability across different kinds of equity: compare the stock of publicly-traded companies with
shares in privately-held businesses. Indeed there are many …nancial assets
that are hardly any less liquid than money, e.g., government bonds. Thus in
our stylized model, "money" should be interpreted very broadly to include
all …nancial assets that are essentially as liquid as money. Under the heading of "equity" come all …nancial assets that are less than perfectly liquid.
By assumption, all these non-monetary assets are subject to the common
resaleability constraint parameterized by .
To understand how …at money can lubricate this economy, notice that
the task of channelling goods from those entrepreneurs who don’t have an
investment opportunity into the hands of those who do is thwarted by the
fact that investors are unable to o¤er savers adequate compensation: the
borrowing constraint ( ) means that new capital investment cannot be entirely self-…nanced by issuing new equity, and the resaleability constraint ( )
means that su¢ cient of the old equity cannot change hands quickly. Fiat
money can help alleviated this problem. Our analysis shows that if and
aren’t high enough –if (and only if) a particular combination of and
lies below a certain threshold – then the circulation of …at money, passing
each period from investors to savers in exchange for goods, serves to boost
aggregate activity. Whenever …at money plays this essential role we say that
the economy is a monetary economy. Whether or not agents use …at money
3
–whether or not the economy is monetary –is determined endogenously.
We show that in a monetary economy, the expected rate of return on
money is very low, less than the expected rate of return on equity. (The
steady-state of an economy where the stock of …at money is …xed would
necessarily have a zero net return on money.) Nevertheless, a saver chooses
to hold some money in his portfolio, because in the event that he has an
opportunity to invest in the future he will be liquidity constrained, and money
is more liquid than equity. The gap between the return on money and the
return on equity is a liquidity premium.
We also show that both the returns on equity and money are lower than
the rate of time preference. This is because borrowing constraints starve the
economy of means of saving –too little equity can be credibly pledged –which
raises asset prices and lowers yields. As a consequence, agents who never
have investment opportunities, such as the workers, choose to hold neither
equity nor money. Assuming workers cannot borrow against their future
labor income, they simply consume their wage, period by period. This may
help explain why certain households neither save nor participate in asset
markets. It isn’t that they don’t have access to those markets, or that they
are particularly impatient, but rather that the return on assets isn’t enough
to attract them. (The model can be extended to show that if workers face
their own liquidity shocks then they may save, but only use money to do so.)
In our - framework, and
are exogenous parameters. Although
the borrowing constraint ( ) and the resaleability constraint ( ) might both
be thought of as varieties of liquidity constraint,1 in this paper we will be
especially concerned with the e¤ects of shocks to , which we identify as
liquidity shocks. We are motivated here by the fact that in the recent …nancial
turmoil many assets –such as asset-backed securities and auction-rate bonds
–that used to be highly liquid became much less resaleable.2 Even though we
will focus on shocks to , it is important to recognize that is an essential
1
Brunnermeier and Pedersen (2009) use "funding liquidity" to refer to the borrowing
constraint and "market liquidity" to refer to the resaleability constraint.
2
In our …rst presentations of this research (see, for example, Kiyotaki and Moore, 2001),
although we separately identi…ed the borrowing and resaleability constraints, for analytical
convenience we set = . However, it helps to keep distinct from , as we do in the
current paper, because we are thus able to pin down the e¤ects of shocks to and identify
a monetary policy that can be used in response.
We made use of the - framework in other papers, though sometimes with di¤erent
notation: Kiyotaki and Moore (2002, 2003, 2005a and 2005b).
4
component of the model. Were to be su¢ ciently close to 1, then new
capital investment could be self-…nanced by issuing new equity and there
would be no need for old equity to circulate (reminiscent of the idea that
in the Arrow-Debreu framework markets need open only once, at an initial
date); liquidity shocks, shocks to , would have no e¤ect.
The mechanism by which liquidity shocks a¤ect our monetary economy
is absent from most real business cycle models. In our model, there is a
critical feedback from asset prices to aggregate activity. Consider a persistent
liquidity shock: suppose falls and is anticipated to recover only slowly. The
impact of this fall in resaleability is to shrink the funds available to investors
to use as downpayment. Further, anticipating lower future resaleability, the
price of equity falls relative to the value of money –think of this as a "‡ight
to liquidity" – which tends to raise the size of the required downpayment
per unit of investment. All in all, via this feedback mechanism, investment
falls as falls. Asset prices and aggregate activity are vulnerable to liquidity
shocks, unlike in a standard general equilibrium asset pricing model.
Our basic model, presented in Sections 2 and 3, has a …xed stock of …at
money. Government is introduced in the full model of Section 4, which examines monetary policy. This model bears some resemblance to a Keynesian
IS-LM model, except that in our model prices and wages are fully ‡exible,
and agents are maximizing.
How might government, through interventions by the central bank, ameliorate the e¤ects of liquidity shocks? Speci…cally, how might policy change
behavior in the private economy?
The central bank can costlessly change the supply of …at money. However,
in our framework with ‡exible prices, a once-and-for-all change in the money
supply by means of a lump-sum transfer to the entrepreneurs –a helicopter
drop –wouldn’t have any e¤ect on aggregate real variables.
The central bank can also buy and hold private equity – albeit that we
impose the same upper bound ( ) on the rate at which it can divest. Unlike
a helicopter drop, an open-market operation to purchase equity by issuing
…at money will a¤ect aggregate activity. We show that the policy works by
shifting up the ratio of the values of money to equity held by the private
sector; cf. Metzler (1951). Investing entrepreneurs are in a position to
invest more when their portfolios are more liquid. In e¤ect, the government
improves liquidity in the private economy by taking relatively illiquid assets
onto its own books. This unconventional form of monetary policy has been
employed by central banks around the world in recent years to ease the
5
…nancial crisis, and appears to have met with some success. Interventions by
the central bank have real e¤ects in our economy because they operate across
a liquidity margin - the di¤erence in liquidity between money and equity; cf.
Tobin (1969).
It is revealing to contrast this countercyclical policy, used to o¤set liquidity shocks, with the procyclical policy our model prescribes for dealing with
productivity shocks. We …nd that a central bank should issue money to
buy equity in high productivity states because it is precisely in those states
that bottlenecks in the …nancial market between savers and investors – the
binding borrowing and resaleability constraints –matter most.
Before we come to this policy analysis, it helps to start with the basic
model without government. We will relate our paper to the literature and
make some …nal remarks in Section 5. Proofs are contained in the Appendix.
2
The Basic Model without Government
Consider an in…nite-horizon, discrete-time economy with four objects traded:
a nondurable output, labor, equity and …at money. Fiat money is intrinsically
useless, and is in …xed supply M in the basic model of this and the next
section.
There are two populations of agents, entrepreneurs and workers, each with
unit measure. Let us start with the entrepreneurs, who are the central actors
in the drama. At date t, a typical entrepreneur has expected discounted
utility
1
X
s t
Et
u(cs )
(1)
s=t
of consumption path fct ; ct+1 ; ct+2 ; :::g, where u(c) = log c and 0 < < 1.
He has no labor endowment. All entrepreneurs have access to a constantreturns-to-scale technology for producing output from capital and labor. An
entrepreneur holding kt capital at the start of period t can employ `t labor
to produce
yt = At (kt ) (`t )1
(2)
output, where 0 <
< 1. Production is completed within the period t,
during which time capital depreciates to kt , 0 < < 1. We assume that
the productivity parameter, At > 0 which is common to all entrepreneurs,
follows a stationary stochastic process. Given that each entrepreneur can
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employ labor at a competitive real wage rate, wt , gross pro…t is proportional
to the capital stock:
yt wt `t = rt kt ;
(3)
where, as we will see, gross pro…t per unit of capital, rt , depends upon
productivity, aggregate capital stock and labor supply.
The entrepreneur may also have an opportunity to produce new capital. Speci…cally, at each date t, with probability
he has access to a
constant-returns technology that produces it units of capital from it units
of output. The arrival of such an investment opportunity is independently
distributed across entrepreneurs and through time, and is independent of
aggregate shocks. Again, investment is completed within the period t – although newly-produced capital does not become available as an input to the
production of output until the following period t+1:
kt+1 = kt + it :
(4)
We assume there is no insurance market against having an investment
opportunity.3 We also make a regularity assumption that the subjective
discount factor is larger than the fraction of capital left after production
(one minus the depreciation rate):
Asssumption 1 :
> :
This mild restriction is not essential, but will make the distribution of capital
and asset holdings across of individual entrepreneurs well-behaved.
In order to …nance the cost of investment, the entrepreneur who has an
investment opportunity can issue equity claims to the future returns from
newly produced capital. Normalize one unit of equity at date t to be a claim
to the future returns from the one unit of investment of date t: it pays rt+1
output at date t+1, rt+2 at date t+2, 2 rt+3 at date t+3, and so on.
We make two critical assumptions. First, the entrepreneur who produces
new capital cannot fully precommit to work with it, even though his speci…c
3
This assumption can be justi…ed in a variety of ways. For example, it may not be
possible to verify that someone has an investment opportunity; or veri…cation may take
so long that the opportunity has gone by the time the claim is paid out. A long-term
insurance contract based on self-reporting does not work here because the people are able
to trade assets covertly. Each of these justi…cations warrants formal modelling. But we
are reasonably con…dent that even if partial insurance were possible our broad conclusions
would still hold. So rather than clutter up the model, we simply assume that no insurance
scheme is feasible.
7
skills will be needed for it to produce output. To capture this lack of commitment power in a simple way, we assume that an investing entrepreneur can
credibly pledge at most a fraction < 1 of the future returns.4 Loosely put,
we are assuming that only a fraction of the new capital can be mortgaged.
We take to be an exogenous parameter: the fraction of new capital
returns that can be issued as equity at the time of investment. The smaller
is , the tighter is the borrowing constraint that an investing entrepreneur
faces. To meet the cost of investment, he has to use any money that he may
hold, and raise further funds by – as far as possible – reselling any holding
of other entrepreneurs’equity that he may have accumulated through past
purchases.
The second critical assumption is that entrepreneurs cannot dispose of
their equity holdings as quickly as money. Again to capture this idea in a
simple way, we assume that, before the investment opportunity disappears,
the investing entrepreneur can resell only a fraction t < 1 of his holding of
other entrepreneurs’equity. (He can use all his money.) This is tantamount
to assuming a peculiar transaction cost per period: zero for the …rst fraction
t of equity sold, and then in…nite.
Like , we take t to be an exogenous parameter: the fraction of equity
holdings that can be resold in each period. The smaller is t , the less liquid
is equity; the tighter is the resaleability constraint.
We suppose that the aggregate productivity At and the liquidity of equity
jointly
follow a stationary Markov process in the neighborhood of the
t
constant unconditional mean (A; ). A shock to At is a productivity shock,
and a shock to t is a liquidity shock. (We do not shock , which is why it
does not have a subscript.)
In general, an entrepreneur has three kinds of asset in his portfolio:
money; his holding of other entrepreneurs’ equity; and the uncommitted
fraction, 1
, of the returns from his own capital, which might loosely be
termed "unmortgaged capital stock" - own capital stock minus own equity
issued.
4
Cf. Hart and Moore (1994), where the borrowing constraint is shown to be a consequence of the fact that the human capital of the agent who is raising funds – here, the
investing entrepreneur –is inalienable.
8
Balance sheet
money holding
own equity issued
holding of other entrepreneurs’equity
own capital stock
net worth
It turns out to be in general hard to analyze aggregate ‡uctuations of
the economy with these three assets, because there is a complex dynamic interaction between the distribution of asset holdings across the entrepreneurs
and their choices of consumption, investment and portfolio. Thus, we make
a simplifying assumption: in every period, we suppose that an entrepreneur
can issue new equity against a fraction t of any uncommitted returns from
his old capital – in loose terms, he can mortgage a fraction t of any asyet-unmortaged capital stock. Think of mortgaging old capital stock – or
reselling equity –as akin to peeling an onion slowly, layer by layer, a fraction
t in each period t.
The upshot of this assumption is that an entrepreneur’s holding of others’
equity and his unmortgaged capital stock are perfect substitutes as means of
saving for him: both pay the same return stream per unit (rt+1 at date t+1,
rt+2 at date t+2, 2 rt+3 at date t+3, ...); and up to a fraction t of both
can be resold/mortgaged per period. In e¤ect, by making the simplifying
assumption we have reduced down to two the number of assets that we need
keep track of: besides money, the holdings of other entrepreneurs’ equity
("outside equity") and the unmortgaged capital stock ("inside equity") can
be lumped together simply as "equity".
Let nt be the equity and mt the money held by an individual entrepreneur
at the start of period t. He faces two "liquidity constraints":
nt+1
(1
)it + (1
mt+1
t)
0:
nt ; and
(5)
(6)
During the period, the entrepreneur who invests it can issue at most it
equity against the new capital. And he can dispose of at most a fraction t of
his equity holding, after depreciation. Inequality (5) brings these constraints
together: his equity holding at the start of period t+1 must be at least 1
times investment plus 1
t times depreciated equity. Inequality (6) says
that his money holding cannot be negative.
9
Let qt be the price of equity in terms of output, the numeraire. qt is also
equal to Tobin’s q: the ratio of the market value of capital to the replacement
cost. Let pt be the price of money. (Warning! pt is customarily de…ned as
the inverse: the price of general output in terms of money. But, a priori,
money may not have value, so better not to make it the numeraire.) The
entrepreneur’s ‡ow of funds constraint at date t is then given by
ct + it + qt (nt+1
it
nt ) + pt (mt+1
m t ) = rt n t :
(7)
The left-hand side (LHS) is his expenditure on consumption, investment
and net purchases of equity and money. The right-hand side (RHS) is his
dividend income, which is proportional to his holding of equity at the start
of this period.
Turn now to the workers. At date t, a typical worker has expected discounted utility
1
X
!
1+
s t
(`0s )
;
(8)
Et
U c0s
1
+
s=t
of consumption path c0t ; c0t+1 ; c0t+2 ; :: given his labor supply path f`0t ; `0t ; `0t ; ::g,
where ! > 0; > 0 and U [ ] is increasing and strictly concave. The ‡ow-offunds constraint of the worker is
c0t + qt (n0t+1
n0t ) + pt (m0t+1
m0t ) = wt `0t + rt n0t :
(9)
The consumption expenditure and net purchase of equity and money in the
LHS is …nanced by wage and dividend income. Workers do not have investment opportunities, and cannot borrow against their future labor income.
n0t+1
0; and m0t+1
0:
(10)
An equilibrium process of prices fpt ; qt ; wt g is such that: entrepreneurs
choose labor demand `t to maximize gross pro…t (3) subject to the production function (2) for a given start-of-period capital stock, and they choose
consumption, investment, capital stock and start-of-next-period equity and
money holdings fct ; it ; kt+1 ; nt+1 ; mt+1 g, to maximize (1) subject to (4) (7); workers choose consumption, labor supply, equity and money holding
c0t ; `0t ; n0t+1 ; m0t+1 to maximize (8) subject to (9) and (10); and the markets
for general output, labor, equity and money all clear.
Before we characterize equilibrium, it helps to clear the decks a little by
suppressing reference to the workers. Given that their population has unit
10
measure, it follows from (8) and (9) that their aggregate labor supply equals
(wt =!)1= . Maximizing the gross pro…t of a typical entrepreneur with capital
kt , we …nd his labor demand, kt [(1
)At =wt ]1= which is proportional to kt :
So if the aggregate stock of capital at the start of date t is Kt , labor-market
clearing requires that
(wt =!)1= = Kt [(1
)At =wt ]1= :
Substituting back the equilibrium wage wt into the LHS of (3), we …nd that
the individual entrepreneur’s maximized gross pro…t equals rt kt where
1
rt = at (Kt )
and the parameters at and
(11)
;
are derived from At ; ; ! and :
1
at =
=
1
+
!
(1 + )
:
+
(At )
1+
+
(12)
Note from (12) that lies between 0 and 1, so that rt –which is parametric
for the individual entrepreneur –declines with the aggregate stock of capital
Kt , because the wage increases with Kt . But for the entrepreneurial sector
as a whole, gross pro…t rt Kt increases with Kt . Also note from (12) that rt is
increasing in the productivity parameter At through at . Later we will show
that in the neighborhood of the steady state monetary equilibrium, a worker
will choose to hold neither equity nor money. That is, the worker simply
consumes his labor income at each date:
c0t = wt `0t :
(13)
We are now in a position to characterize the equilibrium behavior of the
entrepreneurs. Consider an entrepreneur holding equity nt and money mt at
the start of period t. First, suppose he has an investment opportunity: let
this be denoted by a superscript i on his choice of consumption, and startof-next-period equity and money holdings, cit ; nit+1 ; mit+1 . He has two ways
of acquiring equity nit+1 : either produce it at unit cost 1, or buy it in the
market at price qt . (See the LHS of the ‡ow-of-funds constraint (7), where,
recall, it corresponds to investment). If qt is less than 1, the agent will not
11
invest. If qt equals 1, he will be indi¤erent. If qt is greater than 1, he will
invest by selling as much equity as he can subject to the constraint (5). The
entrepreneur’s production choice is similar to Tobin’s q theory of investment.
As the aggregate productivity and liquidity of equity (At ; t ) follow a stochastic process in the neighborhood of constant (A; ), we have the following
claim in the neighborhood of the steady state equilibrium (all the proofs are
in the Appendix):
Claim 1 Suppose that
and
Condition 1 : (1
satisfy
) +
> (1
)(1
):
Then in the neighborhood of the steady state:
(i) the allocation of resources is …rst best;
(ii) Tobin’s q is equal to unity: qt = 1;
(iii) money has no value: pt = 0;
(iv) the gross dividend is roughly equal to the time preference rate plus
.
the depreciation rate: rt ' 1
If the investing entrepreneurs can issue new equity relatively freely and/or
existing equity is relatively liquid –Condition 1 is satis…ed –then the equity
market transfers enough resources from the savers to the investing entrepreneurs to achieve the …rst best allocation.5 There is no advantage to having
investment opportunity; Tobin’s q is equal to 1 (the market value of capital
is equal to the replacement cost) and both investing entrepreneurs and savers
earn the same net rate of return on equity –approximately equal to the time
preference rate. (Note that the usual risk premium is almost negligible in
the …rst best with our logarithmic utility function). Because the economy
achieves the …rst best allocation without money, money has no value.
In the following we want to restrict attention to an equilibrium in which
qt is greater than 1. We also want money to have value in equilibrium. Let
5
In steady state, aggregate saving (which equals aggregate investment) is equal to the
depreciation of capital. The RHS of Condition 1 is the ratio of the aggregate saving of the
(fraction 1
) non-investing entrepreneurs to the aggregate capital stock in …rst best.
The LHS is the ratio of the equity issued/resold by the investing entrepreneurs to the
aggregate capital stock: (1
) corresponds to new equity issued and
corresponds
to old equity resold by the (fraction ) investing entrepreneurs. Thus Condition 1 says that
the equity issued/resold by the investing entrepreneurs is enough to shift the aggregate
saving of the non-investing entrepreneurs.
12
us assume that
and
satisfy:
Assumption 2 : 0 < ( ; ), where
( ; )
2
+[(
[ (1
(1
)(1
)[(1
)(1
) (1
)(1
) + (1
)(1
) (1
)
] [1
) + ( +
)
+
]
(1
)
) ]:
Observe all the terms in the RHS are positive, except for the terms (1
)(1
) (1
)
and (
)(1
) (1
)
. Thus a
su¢ cient condition for Assumption 2 is
(1
) +
<(
)(1
);
< (1
)(1
):
and a necessary condition is
(1
) +
Notice that if Condition 1 in Claim 1 were satis…ed, then this necessary
condition would not satis…ed and there could be no equilibrium with valued
…at money. Under Assumption 2, however, the upper bound on and is
tight enough to ensure that the following claim holds.
Claim 2 Under Assumption 2, in the neighborhood of the steady state:
(i) the price of money, pt , is strictly positive;
(ii) the price of capital, qt , is strictly greater than 1;
(iii) an entrepreneur with an investment opportunity faces the binding
liquidity constraints and will not choose to hold money: mit+1 = 0.
We will be in a position to prove the claim once we have laid out the
equilibrium conditions –we use a method of guess-and-verify in the following.
For completeness, it should be pointed out that for intermediate values of
and which satisfy neither Assumption 2 nor Condition 1, we can show that
money has no value even though the liquidity constraint (5) still binds. To
streamline the paper, we have chosen not to give an exhaustive account of
the equilibria throughout the parameter space.
There is a caveat to Claim 2(i). Fiat money can only be valuable to someone if other people …nd it valuable, hence there is always a non-monetary
equilibrium in which the price of …at money is zero. Thus when there is a
13
]
monetary equilibrium in addition to the non-monetary equilibrium, we restrict attention to the monetary equilibrium: pt > 0. Claim 2(iii) says that
the entrepreneur prefers investment with the maximum leverage to holding
money, even though the return is in the form of equity which at date t+1
is less liquid than money. (Incidentally, even though the investing entrepreneurs don’t want to hold money for liquidity purposes, the non-investing
entrepreneurs do –see below. This is why Claim 2(i) holds).
Thus, for an investing entrepreneur, the liquidity constraints (5) and (6)
are both binding. His ‡ow of funds constraint (7) can be rewritten
cit + (1
qt ) it = (rt +
t qt ) n t
+ pt mt :
(14)
In order to …nance investment it , the entrepreneur issues equity it at price
qt . Thus the second term in the LHS is the investment cost that has to be
…nanced internally: the downpayment for investment. The LHS equals the
total liquidity needs of the investing entrepreneur. The RHS corresponds
to the maximum liquidity supplied from dividends, sales of the resaleable
fraction of equity after depreciation and the value of money. Solving this
‡ow-of-funds constraint with respect to the equity of the next period, we
obtain
R
cit + qtR nit+1 = rt nt + [ t qt + (1
t )qt ] nt + pt mt ;
1
qt
< 1, as qt > 1:
where qtR
1
(15)
(16)
The value of qtR is the e¤ective replacement cost of equity to the investing
entrepreneur: because he needs a downpayment 1
qt for every unit of
investment of which he retains 1
inside equity, he needs (1
qt )=(1
)
to acquire one unit of inside equity. The RHS of (15) is his net worth: gross
dividend plus the value of his depreciated equity nt –of which the resaleable
fraction t is valued at market price and the non-resaleable fraction 1
t is
valued by the e¤ective replacement cost –plus the value of money.
Given the discounted logarithmic preferences (1), the entrepreneur saves
a fraction of his net worth, and consumes a fraction 1
:6
cit = (1
) rt nt + [ t qt + (1
6
R
t )qt ]
n t + pt mt :
(17)
Compare (1) to a Cobb-Douglas utility function, where the expenditure share of
present consumption out of total wealth is constant and equal to 1= 1 + + 2 + ::: =
1
.
14
And so, from (14), we obtain an expression for his investment in period t:
it =
(rt +
t qt ) n t
1
+ pt mt
qt
cit
(18)
:
Investment is equal to the ratio of liquidity available after consumption to
the required downpayment per unit of investment.
Next, suppose the entrepreneur does not have an investment opportunity: denote this by a superscript s to stand for a saver. The ‡ow-of-funds
constraint (7) reduces to
cst + qt nst+1 + pt mst+1 = rt nt + qt nt + pt mt :
(19)
For the moment, let us assume that constraints (5) and (6) do not bind.
Then the RHS of (19) corresponds to the entrepreneur’s net worth. It is the
same as the RHS of (15), except that now his depreciated equity is valued at
the market price, qt . From this net worth he consumes a fraction 1
:
cst = (1
(20)
)(rt nt + qt nt + pt mt ):
Note that consumption of an entrepreneur who does not have investment
opportunity is larger than consumption of an investing entrepreneur if both
hold the same equity and money at the start of period. The remainder is
split across a savings portfolio of mt+1 and nt+1 .
To determine the optimal portfolio, consider the choice of sacri…cing one
unit of consumption ct to purchase either 1=pt units of money or 1=qt units
of equity, which are then used to augment consumption at date t+1. The
…rst-order condition is
u0 (ct ) = Et
pt+1
pt
(1
) u0 cst+1 + u0 (cit+1 )
rt+1 + qt+1 0 s
) Et
u ct+1
qt
(
rt+1 + t+1 qt+1 + 1
+ Et
qt
(21)
= (1
t+1
R
qt+1
u0 cit+1
)
:
The RHS of the …rst line of (21) is the expected gain from holding 1=pt
additional units of money at date t+1: money always yields pt+1 which, proportionately, will increase utility by u0 cst+1 when he does not have a date
15
t+1 investment opportunity (probability 1
) and by u0 (cit+1 ) when he does
(probability ). The second line is the expected gain from holding 1=qt additional units of equity at date t+1. Per unit, this additional equity yields
rt+1 dividend plus its depreciated value. With probability 1
the entrepreneur does not have a date t+1 investment opportunity, the depreciated
equity is valued at the market price, qt+1 , and these yields increase utility in
proportion to u0 cst+1 . With probability the entrepreneur does have an
investment opportunity at date t+1, in which case he will value depreciated
equity by the market price qt+1 for the resaleable fraction and by the e¤ecR
for the non-resaleable fraction, and these yields
tive replacement cost qt+1
increase utility in proportion to u0 cit+1 .
Notice that because the e¤ective replacement cost is lower than the market price, the e¤ective return on equity is lower just when the entrepreneur
is more in need of funds, viz. when an investment opportunity arises and his
marginal utility of consumption is higher (cit+1 < cst+1 ). That is, over and
above aggregate risk, equity carries an idiosyncratic risk: its e¤ective return
is negatively correlated with the idiosyncratic variations in marginal utility
that stem from the stochastic investment opportunities. Money is free from
such idiosyncratic risk.
We are now in a position to consider the aggregate economy. The great
merit of the expressions for an investing entrepreneur’s consumption and
investment choices, cit and it , and a non-investing entrepreneurs’consumption
and savings choices, cst , nt+1 and mt+1 , is that they are all linear in start-ofperiod equity and money holdings nt and mt .7 Hence aggregation is easy: we
do not need to keep track of the distributions. Notice that, because workers
do not choose to save, the aggregate holdings of equity and money of the
entrepreneurs are equal to aggregate capital stock Kt and money supply M .
At the start of date t, a fraction of Kt and M is held by entrepreneurs
who have an investment opportunity. From (18), total investment, It , in new
capital therefore satis…es
(1
qt ) It =
[(rt +
t qt )Kt
+ pt M ]
(1
)(1
t)
qtR Kt : (22)
Goods market clearing requires that total output (net of labor costs,
which equal the consumption of workers), rt Kt , equals investment plus the
7
From (19) and (20), the value of savings, qt nst+1 + pt mst+1 , is linear in nt and mt , and
(the reciprocal of) the portfolio equation (21) is homogeneous in nnt+1 ; mnt+1 , noting that
u0 (c) = 1=c given the logarithmic utility function:
16
consumption of entrepreneurs. Using (17) and (20), we therefore have
rt Kt = at Kt = It + (1
[rt + (1
+
)
t ) qt + (1
(23)
t)
qtR ]Kt
+ pt M :
It remains to …nd the aggregate counterpart to the portfolio equation
(21). During period t, the investing entrepreneurs sell a fraction of their
investment It , together with a fraction t of their depreciated equity holdings
Kt , to the non-investing entrepreneurs. So the stock of equity held by the
group of non-investing entrepreneurs at the end of the period is given by
s
. And, by claim 2(iii), we know that this
It + t Kt + (1
) Kt Nt+1
s
group also hold all the money stock, M . The group’s savings portfolio (Nt+1
,
M ) satis…es (21), which can be simpli…ed to:
(1
=
Et
(rt+1 + qt+1 )=qt pt+1 =pt
s
+ pt+1 M
(rt+1 + qt+1 )Nt+1
R
pt+1 =pt [rt+1 + t+1 qt+1 + (1
t+1 ) qt+1 ]=qt
:
R
s
[rt+1 + t+1 qt+1 + (1
t+1 ) qt+1 ]Nt+1 + pt+1 M
) Et
(24)
Equation (24) lies at the heart of the model. When there is no investment
opportunity at date t+1, so that the partial liquidity of equity doesn’t matter, the return on equity, (rt+1 + qt+1 )=qt ; exceeds the return on money,
pt+1 =pt : the LHS of (24) is positive. However, when there is an investment
opportunity, the e¤ective rate of return on equity, [rt+1 + t+1 qt+1 + (1
R
t+1 ) qt+1 ]=qt ; is less than the return on money: the RHS of (24) is positive.
These return di¤erentials have to be weighted by the respective probabilities
and marginal utilities. Note that, because of the impact of idiosyncratic risk
on the RHS, the liquidity premium of equity over money in the LHS may be
substantial and may vary through time.
Aside from the liquidity shock t and the technology parameter At which
follow an exogenous stationary Markov process, the only state variable in
this system is Kt , which evolves according to
Kt+1 = Kt + It :
(25)
Restricting attention to a stationary price process, the competitive equilibrium can be de…ned recursively as a function (It ; pt ; qt ; Kt+1 ) of the aggregate
state (Kt ; At ; t ) that satis…es (11) ; (22) (25), together with the law of motion of At and t .
17
From these equations it can be seen that there are rich interactions between quantities (It ; Kt+1 ) and asset prices (pt ; qt ). In this sense, our economy
is similar to Keynes (1936): (23) and (24) are akin to IS and LM equations.
In steady state, when at = a (the RHS of (12) with At = A) and t = ,
capital stock K, investment I, and prices p and q, satisfy I = (1
)K and
r+
l= 1
r
(1
+
)l =
)
1
+
+(1
r
(1
)q =
1
1
1
1
(1
(q
(1
(1
1)
r+
)
1
1
) 1
1
1
q)
q
1
1
(26)
(27)
q
q + (l= )
;
+ 1 q + (l= )
(28)
where r = aK 1 ; l = pM=K; and
(1
) + (1
+ ) (the steadystate fraction of equity held by non-investing entrepreneurs at the end of a
period).
Equations (26), (27) and (28) can be viewed as a simultaneous system
in three unknowns: the price of capital, q; the gross pro…t rate on capital,
r; and the value of the money stock as a fraction of total capital, l. (26)
and (27) can be solved for a r and l, each as a¢ ne functions of q, which
when substituted into (28) yield a quadratic equation in q with a unique
positive solution. Assumption 2 is su¢ cient to ensure that this solution lies
strictly above 1 (but below 1= ). We can also show that Assumption 2 is the
necessary and su¢ cient condition for money to have value: p > 0.
As a prelude to the dynamic analysis that we undertake later on, notice
that the technology parameter A only a¤ects the steady-state system through
the gross pro…t term r = aK 1 . That is, a rise in the steady state value of
A increases the capital stock, K, but does not a¤ect q, the price of capital.
The price of money, p, increases to leave l = pM=K unchanged.
It is interesting to compare our economy, in which the liquidity constraints
(5) and (6) bind for investing entrepreneurs, to a "…rst-best" economy without such constraints. Consider steady states. In the …rst-best economy, the
price of capital would equal its cost, 1; and the capital stock, K say, would
equate the return on capital, aK 1 + , to the agents’common subjective
18
return, 1= . (See Claim 1.) We show below that in our constrained economy,
the level of activity – measured by the capital stock K – is strictly below
K . Because of the partial liquidity of equity, the economy fails to transfer
enough resources to the investing entrepreneurs to achieve the …rst-best level
of investment.
On account of the liquidity constraint, there is a wedge between the marginal product of capital and the expected rate(s) of return on equity. It turns
out that the expected rate(s) of return on equity and the rate of return on
money all lie below the time preference. Intuitively, because the rates of return on assets to savers are below their time preference rate, they do not save
enough to escape the liquidity constraint when they have an opportunity to
invest in future.
Claim 3 In the neighborhood of the steady state monetary economy,
(i) the stock of capital, Kt+1 is less than in the …rst-best (unconstrained)
economy:
1
Kt+1 < K , Et at+1 Kt+11 +
> ;
(ii) the expected rate of return on equity (assuming the saver does not
have investment opportunity at date t+1) is lower the time preference rate:
Et
at+1 Kt+11 + qt+1
1
< ;
qt
(iii) the expected rate of return on money is yet lower:
Et
at+1 Kt+11 + qt+1
pt+1
< Et
;
pt
qt
(iv) the expected rate of return on equity contingent on having an investment opportunity in the next period is lower still:
Et
at+1 Kt+11 +
t+1
qt+1 + (1
qt
t+1 )
R
qt+1
<
pt+1
:
pt
Claim 3(iii) follows directly from (28), given that in steady state q > 1.
This di¤erence between the expected return on equity and money, re‡ecting
19
the liquidity premium, equals the nominal interest rate on equity.8
In our monetary economy, there are a spectrum of interest rates. In
descending order: the expected marginal product of capital, the time preference rate, the expected rate of return on equity, the expected rate of return
on money, and the expected rate of return on equity contingent on the saver
having an investment opportunity in the next period. Thus in our economy
the impact of asset markets on aggregate production cannot be summarized
by a single real interest rate as in some popular models such as Woodford
(2003). Equally, it would be misleading to use the rates of returns on money
or equity to calibrate the time preference rate.
The fact that the expected rates of return on equity and money are both
lower than the time preference rate justi…es our earlier assertion that workers
will not choose to save by holding capital or money.9 (Of course, if workers
could borrow against their future labor income they would do so. But we
have ruled this out.) In steady state, workers enjoy a constant consumption
equal to their wages.
The reason why an entrepreneur saves, and workers do not, is because
the entrepreneur is preparing for his next investment opportunity. And the
entrepreneur saves using money as well as equity, despite money’s particularly
low return, because he anticipates that he will be liquidity constrained at
the time of investment. Along a typical time path, he experiences episodes
without investment, during which he consumes part of his saving. As the
return on saving –on both equity and money –is less than his time preference
8
By the Fisher equation, the nominal interest rate on equity equals the net real return
on equity plus the in‡ation rate. But minus the in‡ation rate equals the net real return on
money. Hence the nominal interest rate on equity equals the real return on equity minus
the return on money, i.e. the liquidity premium. Because our money is broad money (all
assets that are as liquid as …at money), our nominal interest rate is akin to the interest
rate in Keynes (1936): the di¤erence in the rate of return on partially liquid assets versus
that on fully liquid assets.
9
Workers may save if they faced their own investment opportunity shocks. Suppose,
for example, that each worker randomly faces a "health shock" which entails immediately
spending some …xed amount in order to maintain his human capital. (Health insurance
may cover some of the cost, but the patient has to make a co-payment from his own
pocket). Then, if the resaleability of equity is low, a worker may choose to save entirely
in money enough to cover the amount . The point is that even though the rate of return
on equity is higher than money, on account of the resaleability constraint he would need
to save more in equity than money, which may be less attractive given that the rate of
return on equity is lower than his time preference rate. See Kiyotaki and Moore (2005a)
for details.
20
rate, the value of his net worth gradually shrinks, as does his consumption.
He only expands again at the time of investment. In the aggregate picture,
we do not see all this …ne grain. But it is important to realize that, even in
steady state, the economy is made up of a myriad of such individual histories.
3
Dynamics and Numerical Examples
In order to examine the dynamics of our economy, let us present numerical examples by specifying a law of motion for productivity and liquidity
(At ; t ) : Suppose that (At ; t ) follows independent AR(1) processes so that
at =
1
1
+
!
(At )
1+
+
(in (12)) and
at
a=
t
=
a
t
(at
1
t 1
follow AR(1) as
a) + "at ;
+ " t;
(29)
(30)
where a and
2 (0; 1) and we set a =
= 0:95 for calibration. The
variables "at and " t are iid innovations of the levels of productivity and
liquidity, which have mean zero and are mutually independent. We present
our numerical examples to illustrate mainly qualitative features of our model.
We follow Del Negro et. al. (2011) for choosing parameters. In particular, we
consider one period is quarterly and use = 0:05 (arrival rate of investment
opportunity), = 0:19 (mortgageable fraction of new investment), = 0:19
(resaleable fraction of equity in the steady state), = 0:4 (share of capital),
= 1 (inverse of Frisch elasticity of labor supply), = 0:99 (utility discount
factor), = 0:975 (one minus depreciation rate). F igure 1 shows the impulse
= 1:43%:
response function to a 1% increase in At , which increases at by 1+
+
21
Figure 1. Impulse Responses of Basic Economy to Productivity Shock
a
I
1 .5
C
2
1 .5
1 .5
%
%
1
%
1
1
0 .5
0 .5
0 .5
0
0
0
20
40
60
80
100
0
0
20
40
K
60
80
100
0
20
40
Y
0 .8
60
80
100
60
80
100
p
1 .5
2
0 .6
1 .5
0 .4
%
%
%
1
1
0 .5
0 .2
0 .5
0
0
0
20
40
60
80
100
60
80
100
0
0
20
40
60
80
100
0
20
40
q
1
%
0 .5
0
-0 . 5
0
20
40
Because capital stock is pre-determined, and the labor market clears, output increases by 1:43% (the same proportion as at ). Then from the goods
market equilibrium condition (23) ; we observe the asset prices (pt ; qt ) have
to increase together with productivity in order to increase consumption and
investment in line with the larger output. Although investment is more
sensitive to the asset prices and thus increases proportionately more than
consumption, the aggregate consumption of both entrepreneurs and workers
increase substantially (especially because workers’ consumption is equal to
their wage income). This is di¤erent from the …rst best allocation under
Condition 1, in which consumption would be much smoother than invest22
ment because, without the binding liquidity constraints, consumption would
depend upon permanent rather than current income. The co-movement of
quantities and asset prices is also a unique features of the monetary equilibrium with binding liquidity constraints. In contrast, in the …rst best Tobin’s
q would always equal 1 and the value of money would always be zero.
Now let us consider liquidity shocks. F igure 2 shows the impulse response
of quantities and asset prices against a 50% fall in the resaleability of the
equity.
Figure 2. Impulse Responses of Basic Economy to Liquidity Shock
φ
I
0
C
0
3
2
-2
-2 0
%
%
%
1
-4
0
-4 0
-6
-6 0
-1
-8
0
20
40
60
80
100
-2
0
20
40
K
60
80
100
0
20
40
Y
0
60
80
100
60
80
100
p
0
40
-0 . 5
30
-0 . 5
%
%
%
-1
20
-1 . 5
-1
10
-2
-2 . 5
0
20
40
60
80
100
60
80
100
-1 . 5
0
0
20
40
60
80
100
0
20
40
q
15
%
10
5
0
0
20
40
When the resaleability of equity falls, and only slowly recovers, the investing entrepreneurs are less able to …nance downpayment from selling their
equity holdings, and so investment decreases substantially. Capital stock and
23
output gradually decrease with persistently lower investment. Also the entrepreneurs without investment opportunities now …nd money more attractive
than equity as a means of saving (holding the rates of return unchanged),
given that they can resale a smaller fraction of their equity holding when
future investment opportunities arise (see (24)). Thus, the value of money
increases compared to the equity price in order to restore asset market equilibrium. This can be thought of as a "‡ight to liquidity": a ‡ight from equity
to money.
Notwithstanding this ‡ight from equity, the equity price tends to rise
with the fall in the liquidity. One way to understand why is to think of
the gap between Tobin’s q and unity as a measure of the tightness of the
liquidity constraint, which increases when the resaleability of equity falls.
Another way is to observe that, because output is not a¤ected initially (given
full employment), consumption must increase to maintain equilibrium in the
goods market; and consumption rises through the wealth e¤ect of a rise
in asset prices. This negative co-movement between investment, asset prices
and consumption is a shortcoming of our basic model –a shortcoming shared
by many macroeconomic model with ‡exible prices.10 We address this in the
next section.
Note that, in contrast to our monetary equilibrium, the …rst best allocation would not react to the liquidity shock as the liquidity constraint would
not be binding.
4
Full Model with Storage and Government
We now present the full model. The negative co-movement in the basic
model between investment, asset prices and consumption can be reversed
by expanding the model to include an alternative liquid means of saving:
storage. Speci…cally, suppose that an agent can store t zt+1 units of goods
at date t to obtain zt+1 units of goods at date t+1, where zt+1 must be
nonnegative. Although the storage technology has constant returns to scale
at the individual level, it is decreasing returns to scale at the society level:
t is an increasing function of the aggregate quantity of storage Zt+1 ;
10
Shi (2011) points out that in our basic model it is di¢ cult for a liquidity shock to
generate a positive co-movement in aggregate investment and the price of equity.
24
t
=
Zt+1
(Zt+1 ) =
; where
0;
> 0:
0
Storage represents all the various means of short-term saving besides money,
such as consumer durables or net foreign asset (domestic residents can save
in foreign assets but cannot borrow from foreigners).
To complete the model, we also introduce government. Our goal here is
simply to explore the e¤ects on equilibrium of an exogenous government policy rule. We make no attempt to explain government behavior. At the start
of date t, suppose the government holds Ntg equity. Unlike entrepreneurs,
the government cannot produce new capital. However, it can engage in open
market operations, to buy (sell) equity by issuing (taking in) money –it has
sole access to a costless money-printing technology. Any sale of equity is
subject to the same constraint as (5).11 Finally, the government can purchase goods, or transfer goods to the workers (a negative would correspond
to a lump-sum tax of the workers). Let Gt denote the total government
purchases. Assume that Gt does not a¤ect the entrepreneurs, which leaves
intact our analysis of their behavior. We assume that Ntg and Gt are not so
large that the private economy switches regimes. That is, we are still in an
equilibrium where the liquidity constraints bind for investing entrepreneurs
and money is valuable.
If Mt is the stock of money privately held by entrepreneurs at the start
of date t, then the government’s ‡ow-of-funds constraint is given by
g
Gt + qt Nt+1
Ntg = rt Ntg + pt (Mt+1
Mt ) = rt Ntg + (
t
1) Lt ; (31)
is the money supply growth
where Lt pt Mt are real balances, and t MMt+1
t
rate. That is, cost of the government’s purchases of output and equity must
be met by the dividends from its equity holding plus seigniorage revenues.
Since government is a large agent, at least relative to each of the private
agents, open market operations will a¤ect the prices pt and qt .
We will suppose that the government follows a rule for its open market
operations and …scal policy:
g
Nt+1
=
K
11
at
a
a
a
+
t
(32)
The government also is subject to the same resaleability constraint as the entrepreneur:
g
(1
t ) Nt .
g
Nt+1
25
Gt = [(rt + qt )Ntg
(Lt
(33)
L)] ;
where a ;
and are policy parameters, and K and L are capital stock and
real money balances in the non-stochastic steady state. The …rst equation
is government’s feedback rule for its open market operations: it chooses the
size of open market operation (ratio of government equity holding to total
equity supply in the steady state) as a function of the proportional deviations
of productivity and liquidity from the steady state levels. This rule implies
that the government’s equity holding is zero in the steady state. The second
equation is the …scal policy rule: the government adjusts its goods purchases
to be proportional to the deviation of its net asset holdings from steady state
at the beginning of each period. To limit the length of our discussion, we
will here report only on simulations with = 0 so that Gt = 0; i.e. where all
the …scal adjustment is done through the money supply growth rate t .
The earlier analysis carries through, with obvious modi…cations. The
total supply of equity (which by construction is equal to the aggregate capital
stock) equals the sum of the government’s holding and the aggregate holding
of the entrepreneurs (denoted by Nt+1 ):
g
Kt+1 = Nt+1
+ Nt+1 :
(34)
Workers consume all their disposable income, and, given the form of their
preferences in (8), government policy does not a¤ect their labor supply.
Equations (22) ; (23) and (24) are modi…ed to:
(1
qt ) It =
[(rt +
t qt )Nt
+ Lt + Zt ]
(1
)(1
t)
qtR Nt
(35)
at Kt + Zt = It + (Zt+1 ) Zt+1 + Gt + (1
)
R
[rt + (1
+ t ) qt + (1
t ) qt ]Nt + Lt + Zt (36)
(1
= Et
) Et
(rt+1 + qt+1 )=qt Lt+1 =( t Lt )
s
(rt+1 + qt+1 )Nt+1
+ Lt+1 + Zt+1
Lt+1 =( t Lt ) [rt+1 +
[rt+1 + t+1 qt+1 + (1
(1
) Et
R
qt+1 + (1
t+1 ) qt+1 ]=qt
R
s
t+1 ) qt+1 ]Nt+1 + Lt+1 + Zt+1
t+1
(rt+1 + qt+1 )=qt (1= (Zt+1 ))
s
(rt+1 + qt+1 )Nt+1
+ Lt+1 + Zt+1
26
(37)
= Et
(1= (Zt+1 )) [rt+1 +
[rt+1 + t+1 qt+1 + (1
R
qt+1 + (1
t+1 ) qt+1 ]=qt
;
s
R
t+1 ) qt+1 ]Nt+1 + Lt+1 + Zt+1
t+1
(38)
g
s
= It + t Nt +(1 ) Nt + Ntg Nt+1
. In the investment equawhere Nt+1
tion, (35), entrepreneurs use their money, storage and the resaleable portion
of their equity –net of their consumption –to …nance the downpayment. In
the goods market equilibrium, (36) ; output (net of the worker’s consumption)
plus storage return equals the sum of investment, new storage, government
purchases and entrepreneurs’consumption. (If storage were considered a net
foreign asset, then the accumulation of net foreign assets, Zt+1 Zt ; would be
the current account.) The portfolio equation (37) gives the trade-o¤ between
holding equity and money. And the new portfolio equation (38) gives the
trade-o¤ between holding equity and storage.
Restricting attention to a stable price process, competitive equilibrium is
g
de…ned recursively as functions It ; rt ; Lt ; qt ; Zt+1 ; Kt+1 ; Nt+1 ; Nt+1
; Gt ; t+1
g
of the aggregate state (Kt ; Zt ; Nt ; at ; t ) that satisfy (11),(25),(31) (38)
together with the exogenous law of motion of (at ; t ) :12
How does the presence of storage as alternative means of liquid saving
storage alter the impulse responses? F igure 3 compares the impulse responses to a liquidity shock in the model without storage (taken from F igure
2) and in the model with storage. We choose a storage technology that has
very close to constant returns to scale ( = 0:0001), and is such that the
steady-state level of storage ( 0 = 0:5) is modest compared to the steadystate capital stock (K = 9:49).
In response to the fall in the resaleability of equity, storage increases
sharply, and investment falls more signi…cantly than the economy without
storage, leading to a more signi…cant fall in output.13 Importantly, consumption can now also fall along with investment, as output is soaked up by the
sharp rise in storage.
Also, because money and storage are very close substitutes –with a rate
12
If there were a lump-sum transfer of money to the entrepreneurs (a helicopter drop),
then aggregate quantities would not change in our economy given that prices and wages are
‡exible. The consumption and investment of individual entrepreneurs would be a¤ected,
however, because there would be some redistribution.
13
In Bernanke and Gertler (1989), this is called "disintermediation": with greater friction in …nancial markets, more funds bypass those markets and are instead channelled
directly into alternative investments –here, into storage –that may be productively inferior.
27
of return very close to one –the price of money is stable. Real balances hardly
increase. As a consequence, the "‡ight to liquidity" induces the equity price
to fall a little, at least initially.
Taking these …ndings together, we see that the presence of an alternative
liquid means of saving has overcome the shortcomings of our basic model.
Quantities (investment and consumption) and asset prices move together, as
storage serves as a bu¤er stock to absorb output and stabilize the value of
money.
Figure 3. Impulse Responses of Economy with Storage to Liquidity Shock
φ
I
0
C
10
4
0
2
%
%
%
-2 0
-1 0
0
-4 0
-2 0
-6 0
0
20
40
60
80
100
-3 0
0
-2
-4
20
40
K
60
80
100
0
20
40
Y
2
1
0
0
60
80
100
60
80
100
Z
100
80
%
%
%
60
-2
-1
40
-4
-2
-6
20
-3
0
20
40
60
80
100
0
0
20
L
40
60
80
100
0
20
q w i t h o u t s t o ra g e
40
40
q w i t h s to ra g e
15
0 .6
30
0 .4
20
%
%
%
10
0 .2
5
10
0
0
0
0
20
40
60
80
100
0
20
40
60
80
100
-0 . 2
0
20
40
60
80
P u re M o n e y
M o n e y & S t o ra g e
How might the government, through its central bank, conduct open market operations in response to the liquidity shock? A …rst-best allocation
28
100
would not be a¤ected by a liquidity shock. With this benchmark in mind, in
our monetary economy the central bank should use open market operations
to o¤set the e¤ects of the liquidity shock, by setting the feedback rule coe¢ cient
to be negative in (32). That is, the central bank should counteract
the negative shock by purchasing equity with money, in order to – at least
partially – restore the liquidity of investing entrepreneurs. F igure 4 compares the impulse responses of the economy without this policy rule ( = 0
and = 0, taken from F igure 3) and with the rule ( = 0:1).
Figure 4. Impulse Responses of Full Model to Liquidity Shock
phi
I
0
C
K
10
1
2
0
0
0
-10
%
%
%
%
-20
-1
-2
-40
-20
-60
0
50
-30
0
100
Y
-2
-3
50
100
-6
0
Z
1
50
100
0
L
100
50
0 .5
50
%
-1
%
%
60
-2
40
0
20
-3
0
0
50
100
0
0
50
100
0
50
100
- 0 .5
0
50
g
N /K
µ
s teady
0 .8
G
100
1
0 .6
0 .5
%
Diff
50
Diff
100
q
80
0
%
-4
0 .4
N o P o lic y
0
W i th P o l i c y
0
0 .2
- 0 .5
0
0
50
100
-50
0
-1
50
29
100
0
50
100
100
The central bank’s purchases of equity with money causes real balance
to increase sharply, notwithstanding the relatively stable price of money.
Storage increases less than in the economy without the policy intervention.
Investment falls initially by 30% –almost as much as in the case of no policy,
because at the time of the shock the investing entrepreneurs’portfolios are
predetermined. However, in the following period, investing entrepreneurs
(most of whom were savers in previous period) have a larger proportion of
liquid assets thanks to the policy intervention, and investment recovers to a
level of 10% below the steady state. Thus capital stock and output do not
fall as much as in the economy without intervention.
After the initial purchase of equity, government runs a surplus because
equity yields a higher return. It uses this surplus to reduce the money
supply by setting t < 1 (assuming no adjustment to government purchases).
Because this de‡ationary policy rewards money holders, the ‡ight to liquidity
is more pronounced: the equity price falls as a result.
In contrast, how might the central bank use open market operations in
response to a productivity shock? Once more taking the …rst-best allocation
as a benchmark, the problem of our laissez-faire monetary economy is that
investment does not react enough to productivity shocks and consumption is
not smooth enough. Here the central bank should provide liquidity procyclically to accommodate productivity shocks, by setting the feedback coe¢ cient
of a to be positive in (32). F igure 5 compares the impulse response functions of the laissez-faire monetary economy with an accommodating monetary policy ( a = 0:2 and = 0). As productivity rises by 1:43%, the central
bank buys equity with money to provide an additional 4% liquidity (again,
notwithstanding the relatively stable price of money). Entrepreneurs hold
more money and less illiquid equity, and thus investment more. Investment
increases by 1:3% in the periods immediately following the shock, rather
than increasing gradually as in the economy without the intervention. But
whereas investment, and hence capital stock and output, all increase more
because of the policy, storage and consumption increase less.
30
Figure 5, Impulse Responses of the Full Model to Productivity Shock
a
I
C
1 .5
1 .5
1 .5
1
1
1
K
0 .8
0 .5
0 .5
%
%
%
%
0 .6
0 .4
0 .5
0 .2
0
0
0
50
100
0
0
Y
50
100
0
0
Z
50
100
0
L
1 .5
6
6
1
4
4
50
100
q
0 .6
0 .5
2
%
%
%
%
0 .4
0 .2
2
0
0
0
0
50
100
0
0
50
100
0
50
100
-0 . 2
0
50
g
-3
x 10
N /K
µ
s teady
3
4
2
2
G
1
Diff
%
Diff
0 .5
1
N o P o lic y
0
W i th P o l i c y
0
-0 . 5
0
-2
0
50
100
-1
0
50
100
0
50
100
The e¢ cacy of these open market operations relies on the purchase of an
asset –here, equity –which is only partially resaleable and hence earns a nontrivial liquidity premium. If the liquidity premium of short-term government
bonds is very low (as in Japan since the late 1990s), then traditional open
market operations will only serve to change the composition of broad money
and will have limited e¤ects. The recent unorthodox policy of the Federal
Reserve Bank (and the Bank of England), such as the Term Security Lending
Facility, is an attempt to increase liquidity by supplying treasury bills against
only partially resaleable securities, such as mortgage backed securities.
31
100
5
Related Literature and Final Remarks
We hope to have succeeded in constructing a model of money and liquidity
in the tradition of Keynes (1936) and Tobin (1969). The two key equations
of our model, (23) and (24) –which are generalized in (36) ; (37) and (38) –
have the ‡avor of the Keynesian IS-LM system. We follow Tobin in placing
emphasis on the spectrum of liquidity across di¤erent classes of asset. Also,
Tobin’s q-theory …nds echo in our model through the central role played by
the equity price q: driving the feedback from asset markets to the rest of the
economy. Our policy prescriptions – use open market operations to change
the liquidity mix of the private sector’s asset holdings – parallel those in
Metzler (1951). Perhaps, with its focus on liquidity, our framework harks
back to an earlier tradition of interpreting Keynes, and has less in common
with the formal Keynesian literature, with its emphasis on sticky prices, that
has been dominant in the past few decades.
This paper is part of the recent literature on macroeconomics with …nancial frictions, that includes Bernanke and Gertler (1989), Kiyotaki and Moore
(1997), Holmstrom and Tirole (1998).14 Naturally, the common thread of
this literature has been some form of borrowing constraint, akin to our constraint. Our innovation here is to combine it with the -constraint, the
resaleability constraint. We have shown that the presence of these two constraints opens up the possibility for …at money to circulate, to lubricate the
transfer of goods from savers to investors. There is a wedge between money
and other assets, that arises out of the assumed di¤erence in their resaleability.
Wedges between assets can be generated in other ways. In limited participation models, agents may have di¤erent access to asset markets.15 Models
with spatially separated markets –island models –assume that agents cannot visit all markets within the period, which limits trade across assets.
Some models combine geographical separation with asynchronization, where
agents have access to asset markets at di¤erent times.16 If the assumption
of competitive markets is dropped, as in matching models, assets can ex14
Surveys can be found in Bernanke, Gertler and Gilchrist (1999), Gertler and Kiyotaki
(2010), Brunnermeier, Eisenbach and Sannikov (2011).
15
See, for example, Allen and Gale (1994, 2007).
16
See, for example, Townsend (1987), Townsend and Wallace (1987), Freeman (1996a,
1996b), and Green (1999).
32
hibit di¤erent degrees of resaleability.17 And there is a long tradition in the
banking and …nance literature that, implicitly or explicitly, has to do with
the limited resaleability of securities, dating back at least to Diamond and
Dybvig (1983).18
We should end by stressing that if, in particular, our model is to be used
for proper policy analysis then considerably more research is still needed.
While it might be argued that our - framework has the virtue of simplicity, as they stand the borrowing and resaleability constraints are too stylized
in nature, too reduced-form. The borrowing constraint can be rationalized
by invoking a moral hazard argument, viz., to produce future output from
new capital requires the speci…c skill of the investing entrepreneur, and he can
renege on his promises. But the resaleability constraint requires more modelling, not least because we need to understand where the liquidity shocks,
the shocks to , come from.19 Can policies be devised that directly dampen
these shocks (or even raise the average value of ), rather than merely dealing
with their e¤ects?
To analyze the e¤ects of open market operations over the business cycle,
we assumed that the government can commit to a policy. But can it? This
question calls for further modelling too, because if the government could
commit to, say, a de‡ationary monetary policy that followed the Friedman
rule (set the real return on money to equal agents’subjective discount rate),
then it would in e¤ect be using its taxation powers to substitute perfect
public commitment for imperfect private commitment. In the long run, can
the government be trusted more than the private sector? And to what extent
do future tax liabilities crowd out a private agent’s ability to issue credible
promises to others?20 These thorny issues warrant much careful thinking.
17
Matching models that can be easily used for policy analysis include Shi (1997), Lagos
and Wright (2005), and Nosal and Rocheteau (2011).
18
For attempts to incorporate banking into standard business cycle models, see, for
example, Wiiliamson (1987), Gertler and Kiyotaki (2010), Gertler and Karadi (2011).
19
Kiyotaki and Moore (2003) shows how the resaleability constraint can arise endogenously due to adverse selection and how securitization may mitigate the adverse selection.
Other macroeconomic models of adverse selection in asset markets inclde Eisfeldt (2004),
Moore (2010), Kurlat (2011).
20
A related question would be: If the government has a superior power to force private
agents to pay, why doesn’t it provide them with …nance directly?
33
6
References
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and Volatility of Asset Prices." American Economic Review 84(4): 933-955.
_______ and ______ 2007. Understanding Financial Crises. Oxford: Oxford University Press.
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Business Fluctuations," American Economic Review 79(1): 14-31.
Bernanke, Ben, Mark Gertler and Simon Gilchrist. 1999. "The Financial
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Brunnermeier, Markus, and Lasse Pedersen. 2009. "Market Liquidity
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_______. 1996b. "Payment System, Liquidity, and Rediscounting."
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Gertler, Mark, and Peter Karadi. 2011. "A Model of Unconventional
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Hart, Oliver, and John Moore. 1994. "A Theory of Debt Based on the
Inalienability of Human Capital." Quarterly Journal of Economics 109(4):
34
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Holmstrom, Bengt. and Jean Tirole. 1998. "Private and Public Supply
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Kiyotaki, Nobuhiro, and John Moore. 1997. "Credit Cycles." Journal of
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_______ and _______. 2001. "Liquidity, Business Cycles and
Monetary Policy: Manuscript of Clarendon Lecture 2." Mimeo. London
School of Economics.
_______ and _______. 2002. "Evil is the Root of All Money."
American Economic Review: Papers and Proceedings 92(2): 62-66.
_______ and _______. 2003. "Inside Money and Liquidity."
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_______ and _______. 2005a. "Liquidity and Asset Prices."
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_______ and _______. 2005b. "Financial Deepening." Journal
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Kurlat, Pablo. 2011. "Lemons Markets and the Transmission of Aggregate Shocks." Mimeo. Stanford University.
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463-484.
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Nosal, Ed, and Guillaume Rocheteau. 2011. Money, Payments, and
Liquidity. Cambridge: MA. MIT Press.
Shi, Shouyong. 1997. "A Divisible Search Model of Fiat Money," Econometrica 65(1), 75-102.
_______. 2011. "Liquidity, Assets and Business Cycles." Mimeo,
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35
_______ and Neil Wallace. 1987. "A Model of Circulating Private
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36
7
7.1
Appendix
Proof of Claim 1
We construct a competitive equilibrium which satis…es Claim 1 under Condition 1. Suppose that inequalities (5) ; (6) and (10) are not binding. Then
from (7) ; we need qt = 1: (If qt were strictly larger than one, investing entrepreneurs would invest arbitrary large amount and (5) would be binding. If qt
were strictly smaller than one, investing entrepreneurs would not invest at all,
which is not consistent with the equilibrium with positive gross investment
in the neighborhood of the steady state). Then the choice of entrepreneurs
and workers in the competitive equilibrium imply:
1 = Et
rt =
At
wt = (1
At (Kt ) (`t )1
ct
ct+1
`t
Kt
)At
(rt+1 + ) = Et
c0t
c0t+1
(rt+1 + ) ; (A1)
1
Kt
`t
= Ct + c0t + Kt+1
= !(`t )
Kt ;
where Kt is aggregate capital stock, `t is aggregate labor, and Ct and c0t are
aggregate consumption of entrepreneurs and workers. Because these are the
conditions for the …rst best allocation, the competitive equilibrium achieves
the …rst best allocation if (5) ; (6) and (10) are not binding. In this competitive equilibrium, we get pt = 0: (If pt were strictly positive with non-binding
ct pt+1
which would not be satis…ed in the
(6) ; we would have 1 = Et
ct+1 pt
neighborhood of the steady state).
Consider this competitive economy in which workers do not save so that
0
ct = wt `t and aggregate investment is equal to aggregate saving of entrepreneurs:
It = Kt+1
Kt = rt Kt Ct :
Using (7) with qt = 1 and pt = 0; the inequality (5) for the investing entrepreneur becomes
rt n t
ct = nt+1
nt
37
(1
)it
t
nt :
Aggregating this inequality for all the investing entrepreneurs, observing the
arrival of the investment opportunity is iid. across entrepreneurs and over
time, we have
(rt Kt
Ct ) = It
(1
)It
t
(A2)
Kt :
In the steady state, Condition 1 implies
(1
)K > (1
)(1
)K
K:
Then in the neighborhood of the steady state (in which I = (1
)K),
inequality (A2) is satis…ed. Because pt = 0; (6) and the second inequality of
(10) are not binding. Therefore under Condition 1, we can …nd a competitive
equilibrium in which the inequalities (5) ; (6) and (10) are not binding and
c0t ' wt `t and n0t ' 0 in the neighborhood of the steady state. Claim 1(iv)
follows from (A1). QED:
7.2
Derivation of Consumption and Portfolio Equations
Let Vt (mt ; nt ) be the value function of the entrepreneur who holds money and
equity (mt ; nt ) at the beginning of the period t before meeting an opportunity
to invest with probability . The Bellman equation can be written as
Vt (mt ; nt ) =
M ax
cit ; it ; mit+1 ; nit+1
ln cit + Et Vt (mit+1 ; nit+1 )
st:(5)(6)(7)
+ (1
)
M ax
cst ; mst+1 ; nst+1
ln cst + Et Vt (mst+1 ; nst+1 )
st:(5)(6)(7), it = 0
Solving the ‡ow-of-funds condition (14) and (15) for consumption cst and cit ,
the Bellman equation is
Vt (mt ; nt )
=
M
axi
i
ln [rt +
mt+1 ;nt+1
+ (1
)
M
axs
s
mt+1 ;nt+1
t
R
qt + (1
qtR nit+1
t ) qt ]nt + pt mt
+ Et Vt (mit+1 ; nit+1 )
ln (rt + qt ) nt + pt mt qt nst+1
+ Et Vt (mst+1 ; nst+1 )
38
pt mst+1
pt mit+1
:
:
Let Rmt+1 be the rate of return on money from date t to date t+1 and
hh0
let Rt+1
be the implied rate of returns on equity for the entrepreneur when
her type is h (h = i for investing and h = s for saving or non-investing) at
date t and h0 at date t+1, i.e.,
pt+1
;
pt
rt+1 +
rt+1 + qt+1
si
=
; Rt+1
=
qt
rt+1 +
rt+1 + qt+1
ii
=
=
;
R
t+1
qtR
Rmt+1 =
ss
Rt+1
is
Rt+1
t+1 qt+1
+
qt
t+1 qt+1 +
qtR
1
t+1
R
qt+1
1
t+1
R
qt+1
;
:
Then the …rst order conditions are
1 = Et
1 > Et
1 = Et
1 = Et
cit
ii
Rt+1
+ (1
ciit+1
cit
Rmt+1 + (1
ciit+1
cst si
Rt+1 + (1
csi
t+1
cst
Rmt+1 + (1
csi
t+1
cit
is
Rt+1
cis
t+1
ci
) is t Rmt+1
ct+1
cs ss
) ss t Rt+1
ct+1
cs
) ss t Rmt+1
ct+1
)
(A3)
(A4)
(A5)
(A6)
0
where cht is date-t consumption of the entrepreneur of type h and chh
t+1 is
date t+1 consumption of entrepreneur when her type is h at date t and h0
at date t+1.
We guess that
R
) [rt + t qt + (1
cit = (1
t ) qt ]nt + pt mt ;
cst = (1
) f(rt + qt )nt + pt mt g ;
i
R
R
nt+1 =
[rt + t qt + (1
t ) qt ]nt + pt mt =qt ;
mit+1 = 0;
R
nst+1 = ft [rt + t qt + (1
t ) qt ]nt + pt mt =qt ;
mst+1 =
(1
ft ) [rt +
t
qt + (1
39
t)
qtR ]nt + pt mt =pt
ciit+1
cis
t+1
csi
t+1
css
t+1
=
=
=
=
(1
(1
(1
(1
R i
) [rt+1 + t+1 qt+1 + (1
t+1 ) qt ]nt+1 ;
) (rt+1 + qt+1 )nit+1 ;
R s
s
) [rt+1 + t+1 qt+1 + (1
t+1 ) qt ]nt+1 + pt+1 mt+1 ;
) (rt+1 + qt+1 )nst+1 + pt+1 mst+1 :
where ft qt nst+1 =(pt mst+1 + qt nst+1 ) 2 (0; 1) is the share of equity in portfolio
of the non-investing entrepreneur. Under this guess we learn
ciit+1
cis
t+1
ii
is
=
R
;
and
= Rt+1
;
t+1
cit
cit
and thus (A3) is always satis…ed. (A4) can be written as
1 > Et
Rmt+1
+ (1
ii
Rt+1
)
Rmt+1
:
is
Rt+1
(A7)
We also learn that
csi
t+1
=
cst
css
t+1
=
cst
si
ft Rt+1
+ (1
ft )Rmt+1 ;
ss
ft Rt+1
+ (1
ft )Rmt+1 :
Thus (A5) and (A6) become
1 = Et
1 = Et
si
Rt+1
+ (1
si
ft Rt+1
+ (1 ft )Rmt+1
Rmt+1
+ (1
si
ft Rt+1 + (1 ft )Rmt+1
ss
Rt+1
(A8)
;
ss
ft Rt+1
+ (1 ft )Rmt+1
Rmt+1
)
(A9)
:
ss
ft Rt+1 + (1 ft )Rmt+1
)
Then ft times the RHS of (A8) plus 1 ft times RHS of (A9) is always equal
to 1, and one of (A8) and (A9) is not independent. Subtracting (A9) from
(A8) side by side and arranging, we get
Et
si
Rmt+1 Rnt+1
= (1
si
ft Rt+1
+ (1 ft )Rmt+1
)
ss
Rt+1
Rmt+1
: (A10)
ss
ft Rt+1 + (1 ft )Rmt+1
This is equivalent to (24) in the text. Because qt > 1 > qtR in our equilibrium,
we always have
ii
si
is
ss
> Rt+1
:
Rt+1
> Rt+1
and Rt+1
40
In the neighborhood of the steady state equilibrium, we have
ii
ss
Rt+1
> Rt+1
> Rmt+1 :
Thus, comparing (A9) and (A7) ; we learn the inequality (A7) holds in the
neighborhood of the steady state.
7.3
Proof of Claim 2
Let l be the ratio of real money balance to capital stock in the steady state.
From (26; 27; 28) in the text, the steady state value of (r; q; l) solves
r=1
(1
+ (1
) (1
(1
) r + (1
+
q) = [ (r +
)
r
q
+
1
(r + q) + l
)q +
q)
(1
=
[r +
1
where
Because
we have
(1
=
1
1
,
)
r
l
=1
+
r = (1
+
l = (1
)
( +
=
1
1
1
1
+
h
( +
(1
(1
( +
)(1
(A12)
)q R
(A13)
)q R ] + l
q
as in the text.
h(1
b+
) and b = (1
h
) + (1
)
(1
(1
) +b =
l > 0; i¤ q <
q+ (1
q
q + (1
where = (1
) + (1
+ ) and q R =
From (A11) and (A12) ; we have
(A11)
) q R + l];
) (1
r+
1
) qR + l ;
(1
)(1
(1
+
)
)
) +b
)
!
q
) : Thus
i
b q
(A14)
i
) + b q:
) +
)i
+ (1
qb:
(A15)
) > 0;
(A16)
Then there is a monetary equilibrium with …nancing constraint only if qb > 1;
or
(1
) +
< (1
) (1
);
41
i.e., Condition 1 in the text for the …rst best allocation is violated.
(A13) can be written as
[r
(1
)q] r
(1
)q + q +
l
=
(q 1) r
(1
)q +
q+
l
(A17)
Together with (A14) and (A15) ; we have the condition for steady state q as
0 =
(q)
0 n
(1
1) @
(q
n
(1
0 n
(1
@
+
+
+
) + q (1
+
h
) + q (1
h
) + q (1
)
+ (1
)
(1
)
+
b
h
)
(1
)
b
)
Then we learn
(1) =
h
+q
h
b
+q
io
b
b
io h
i
b
io h
i
i 1
b
)+
A
(1
i 1
b
)+
A:
(1
(A18)
2
f(1
) [ (1
)(1
) + ] + (1
)g < 0:
Therefore the necessary and su¢ cient condition for the existence of monetary equilibrium with q 2 (1; qb) is
0<
(A19)
(b
q) :
Using (A16), (A19) becomes
0 <
=
2
(1
(1
)[
) (1
Multiplying both sides with 1
0 <
2
(1
+[(
[ (1
(1
)
)+
[
(1
(1
)] [
)] (1
(1
+
)]
)[
(1
; we get the condition
)(1
)[(1
)(1
) (1
)(1
) (1
)
] [(1
) (1
) + (1
)+ ( +
42
)
)(1
]
)+
)] :
(1
)]
) (1
)] :
This is equivalent to Assumption 2 in the text. Therefore, under Assumption
2, we have a competitive equilibrium in which …at money has a positive value
(Claim 3(i) pt > 0) and qt > 1 (Claim 2(ii)). Claim 2(iii) directly follows
from inequality (A4) or (A7) which we proved above given that qt > 1 in the
neighborhood of the steady state. QED:
7.4
Proof of Claim 3
Claim 3(i):
Under Assumption 2, we have (1
have
) +
@ RHS of (A11)
=
@q
(1
) 1
>
(1
)
Given q > 1 and l
< (1
)(1
1
1
+
(1
): Thus we
)(1
)
> 0:
0; we have from (A11)
r>1
)(r + ); or
+ (1
r+
>
1
: QED:
Claim 3(iv):
Suppose that 3(iv) is not true. From (A13) and (A11) respectively, we
have
r
1
q
and
r
r
1
(1
) (1
) (1
) + (1
)+
+
l:
q
q
q
Together these imply l < 0: This is contradiction. QED:
Claim 3(iii):
From equation (A13), Claim 3(iv) implies
Claim 3(ii):
43
r
q
+
> 1: QED:
Using q R < 1 < q in (A13) ; we have
(1
) [r
(1
r
)q] >
(1
q
r
q
)q >
(q
) q R ; or
(1
1) > 0:
Hence, given l > 0; from (A17) it follows that
(
r
q
where
+
1
=
q
(1
+ 1)
<
+
1
q
1
;
1 > 0 by Claim 3(iii). But from (A14) ;
n
1
( +1
) (1
)[
(1
+ )
)
Substituting this to the above inequality, we get
(
(
< 1
+ 1)
+
+ ) (1
+ )
n
1
(1
(1
+ )
)
(1
The LHS of this inequality is increasing in
Suppose Claim 3(ii) is not true, i.e.,
1
(1
< 1
+
1
+
(1
(1
+
) (1
n
(1
+
(1
<
) [1
(1
+
2 2 2
+
(1
) [1
)(1
)
)
:
1
o
b] :
: Then
)
)
Multiplying this inequality through by
(1
(1
)
+
)
)[
o
b] :
o
b] :
(1
)[
(1
(1
+
) ; we have
) ]+
(1
+ (1
) ]
)
(1
)(1
)
2
+
(1
)
) :
Cancel the two terms which do not have factor 1
; and divide by 1
we get
(1
+
) [1
(1
)] < 0; or
(1
+
) [1
(1
This is contradiction. QED:
44
)] < 0:
;